Manhattan-Geodesic Embedding of Planar Graphs

  • Bastian Katz
  • Marcus Krug
  • Ignaz Rutter
  • Alexander Wolff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)


In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is \(\mathcal{NP}\)-hard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is \(\mathcal{NP}\)-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.


Planar Graph Hamiltonian Cycle Hamiltonian Path Point Correspondence Left Endpoint 
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  1. 1.
    Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. J. Graph Algorithms Appl. 10(2), 353–363 (2006)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Demaine, E.: Simple polygonizations (2007), (Accessed May 30, 2009)
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  4. 4.
    Goaoc, X., Kratochvíl, J., Okamoto, Y., Shin, C.-S., Spillner, A., Wolff, A.: Untangling a planar graph. Discrete Comput. Geom (2009),
  5. 5.
    Hurtado, F.: Personal communication (2006)Google Scholar
  6. 6.
    Katz, B., Krug, M., Rutter, I., Wolff, A.: Manhattan-geodesic point-set embeddability and polygonization. Technical Report 2009-17, Universität Karlsruhe (2009),
  7. 7.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Liu, Y., Marchioro, P., Petreschi, R., Simeone, B.: Theoretical results on at most 1-bend embeddability of graphs. Acta Math. Appl. Sinica (English Ser.) 8(2), 188–192 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    O’Rourke, J.: Uniqueness of orthogonal connect-the-dots. In: Toussaint, G. (ed.) Computational Morphology, pp. 97–104. North-Holland, Amsterdam (1988)Google Scholar
  10. 10.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graph. Combinator. 17(4), 717–728 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Raghavan, R., Cohoon, J., Sahni, S.: Single bend wiring. J. Algorithms 7(2), 232–257 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rappaport, D.: On the complexity of computing orthogonal polygons from a set of points. Technical Report SOCS-86.9, McGill University, Montréal (1986)Google Scholar
  13. 13.
    Rendl, F., Woeginger, G.: Reconstructing sets of orthogonal line segments in the plane. Discrete Math 119(1-3), 167–174 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Symp. on Discrete Algorithms (SODA 1990), pp. 138–148 (1990)Google Scholar
  15. 15.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Bastian Katz
    • 1
  • Marcus Krug
    • 1
  • Ignaz Rutter
    • 1
  • Alexander Wolff
    • 2
  1. 1.Faculty of InformaticsUniversität Karlsruhe (TH), KITGermany
  2. 2.Institut für InformatikUniversität WürzburgGermany

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